This chapter determines how the systems behave with time whensubjected to some disturbance. E.g. A microprocessor switches on amotor. The speed will not attain immediately but it will take sometime to attain full speed.In order to understand the behavior of the systems,mathematical models are needed. These models are equationswhich describe the relationship between the input and output of asystem. The basis for any mathematical model is provided by thefundamental physical laws that govern the behavior of the system. Inthis chapter a range of systems will be considered includingmechanical, electrical, thermal & fluid examples.Systems can be made up from a range of building blocks from anumber of basic building blocks.

MECHANICAL SYSTEM BUILDING BLOCKS

The basic building blocks of the models used to representmechanical systems are1) Springs 2) dashpots3) masses

Springs

Springs represents the stiffness of the system. The fig. shows aspring subjected to force F.

1

In case of spring the extension (or) compression is proportional tothe applied forces.

x K F

.

=

F – Applied forcex – extension k – a constantThe spring when stretched stores energy, the energy beingreleased when the spring back to its original length. The energystored,

K F x K E

2.21

22

==

Dash Pots

Dashpots building blocks represent the types of forcesexperienced when we push the object through a fluid or move anobject against frictional forces.In ideal case damping or resisting force F is proportional to thevelocity of the piston. Thus

F = C v

V – Velocity of piston C – a constant

dt dxC F

=

(Since velocity is the rate of change of displacement x.)

Masses

2

Masses represent the inertia or resistance to acceleration.According to Newton’s II law

F = ma

dt dvm

=

=

22

dt xd m

There is also energy stored in mass, when it is moving withvelocity V

1

. The energy being referred to as kinetic energy, andreleased when it stops moving.

2

21

mv E

×=

However there is no energy stored in the dashpot. It does notreturn to the original position, when there is no force input. Thedashpot dissipates energy rather than spring. The power dissipateddepending on the velocity V and being given by.

P = C V

2

ROTATIONAL SYSTEMS

The spring, dashpot and mass are the basic building blocks for mechanical systems when forces and straight line displacementsare involved without any rotation.If there is rotation then the equivalent three building blocks are a